On the computation of the third order terms of the series defining the center manifold for a scalar delay differential equation (Q431111)

Consider the scalar delay differential equation \[ \dot x(t)= ax(t)+ bx(t- r)+ f(x(t), x(t- r)) \] with \(a,b\in\mathbb{R}\), \(r> 0\), \[ f(x,y)= \sum_{j,k\geq 0,j+k\geq 2} {1\over j! k!} c_{j,k} x^j y^k. \] Assume that the equation \(\lambda- a- e^{-\lambda r}b= 0\) has a pair of pure imaginary solutions \(\lambda_{1,2}= \pm i\omega\), \(\omega> 0\), and all other eigenvalues have negative real part. Let \({\mathcal L}:= \{\psi\in C([-r,0],\mathbb{R})\}\), \({\mathcal L}_c:={\mathcal L}+ i{\mathcal L}\). The eigenfunctions to \(\lambda_{1,2}\) are given by \(\varphi_{1,2}(s):= e^{\pm i\omega s}\in{\mathcal L}_c\) and they span the subspace \(\mathcal M\) of \(\mathcal L_c\). One obtains \(\mathcal L_c=\mathcal M+\mathcal N\) using the projector \(P:\mathcal L_c\to\mathcal M\) such that \(I-P:\mathcal L_c\to\mathcal N\). Under these assumptions, there is a local invariant manifold, called the center manifold, which is smooth and tangent to \(\mathcal M\) at the origin, and is the graph of a function \(w\) defined on a neighborhood of the origin in \(\mathcal M\) and taking values in \(\mathcal N\). \(w\) can be represented in the form \[ w(z,\overline z)= \sum_{i+ j \geq 2} \frac {1} {i!j!} w_{i,j} z^i\overline z^j, \] where \(w_{i,j}\in \mathcal L_c\). To study non-degenerate Hopf bifurcation, only the functions \(w_{i,j}\) with \(i+ j=2\) are needed. In the case of degenerate Hopf bifurcation (including Bautin bifurcation) also the function \(w_{2,1}\) is needed. The author shows how to overcome the difficulties in determining \(w_{2,1}\) uniquely. Especially a formula is provided for \(w_{2,1}(0)\).